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Data Mining Cluster Analysis: Basic Concepts and Algorithms

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Title: Data Mining Cluster Analysis: Basic Concepts and Algorithms


1
Data MiningCluster Analysis Basic Concepts and
Algorithms
  • Lecture Notes for Chapter 8
  • Introduction to Data Mining
  • by
  • Tan, Steinbach, Kumar

10/30/2007 Introduction to Data Mining
1
2
What is Cluster Analysis?
  • Finding groups of objects such that the objects
    in a group will be similar (or related) to one
    another and different from (or unrelated to) the
    objects in other groups

3
Applications of Cluster Analysis
  • Understanding
  • Group related documents for browsing, group genes
    and proteins that have similar functionality, or
    group stocks with similar price fluctuations
  • Summarization
  • Reduce the size of large data sets

Clustering precipitation in Australia
4
What is not Cluster Analysis?
  • Simple segmentation
  • Dividing students into different registration
    groups alphabetically, by last name
  • Results of a query
  • Groupings are a result of an external
    specification
  • Clustering is a grouping of objects based on the
    data
  • Supervised classification
  • Have class label information
  • Association Analysis
  • Local vs. global connections

5
Notion of a Cluster can be Ambiguous
6
Types of Clusterings
  • A clustering is a set of clusters
  • Important distinction between hierarchical and
    partitional sets of clusters
  • Partitional Clustering
  • A division data objects into non-overlapping
    subsets (clusters) such that each data object is
    in exactly one subset
  • Hierarchical clustering
  • A set of nested clusters organized as a
    hierarchical tree

7
Partitional Clustering
Original Points
8
Hierarchical Clustering
Traditional Hierarchical Clustering
Traditional Dendrogram
Non-traditional Hierarchical Clustering
Non-traditional Dendrogram
9
Other Distinctions Between Sets of Clusters
  • Exclusive versus non-exclusive
  • In non-exclusive clusterings, points may belong
    to multiple clusters.
  • Can represent multiple classes or border points
  • Fuzzy versus non-fuzzy
  • In fuzzy clustering, a point belongs to every
    cluster with some weight between 0 and 1
  • Weights must sum to 1
  • Probabilistic clustering has similar
    characteristics
  • Partial versus complete
  • In some cases, we only want to cluster some of
    the data
  • Heterogeneous versus homogeneous
  • Clusters of widely different sizes, shapes, and
    densities

10
Types of Clusters
  • Well-separated clusters
  • Center-based clusters
  • Contiguous clusters
  • Density-based clusters
  • Property or Conceptual
  • Described by an Objective Function

11
Types of Clusters Well-Separated
  • Well-Separated Clusters
  • A cluster is a set of points such that any point
    in a cluster is closer (or more similar) to every
    other point in the cluster than to any point not
    in the cluster.

3 well-separated clusters
12
Types of Clusters Center-Based
  • Center-based
  • A cluster is a set of objects such that an
    object in a cluster is closer (more similar) to
    the center of a cluster, than to the center of
    any other cluster
  • The center of a cluster is often a centroid, the
    average of all the points in the cluster, or a
    medoid, the most representative point of a
    cluster

4 center-based clusters
13
Types of Clusters Contiguity-Based
  • Contiguous Cluster (Nearest neighbor or
    Transitive)
  • A cluster is a set of points such that a point in
    a cluster is closer (or more similar) to one or
    more other points in the cluster than to any
    point not in the cluster.

8 contiguous clusters
14
Types of Clusters Density-Based
  • Density-based
  • A cluster is a dense region of points, which is
    separated by low-density regions, from other
    regions of high density.
  • Used when the clusters are irregular or
    intertwined, and when noise and outliers are
    present.

6 density-based clusters
15
Types of Clusters Conceptual Clusters
  • Shared Property or Conceptual Clusters
  • Finds clusters that share some common property or
    represent a particular concept.
  • .

2 Overlapping Circles
16
Types of Clusters Objective Function
  • Clusters Defined by an Objective Function
  • Finds clusters that minimize or maximize an
    objective function.
  • Enumerate all possible ways of dividing the
    points into clusters and evaluate the goodness'
    of each potential set of clusters by using the
    given objective function. (NP Hard)
  • Can have global or local objectives.
  • Hierarchical clustering algorithms typically
    have local objectives
  • Partitional algorithms typically have global
    objectives
  • A variation of the global objective function
    approach is to fit the data to a parameterized
    model.
  • Parameters for the model are determined from the
    data.
  • Mixture models assume that the data is a
    mixture' of a number of statistical
    distributions.

17
Map Clustering Problem to a Different Problem
  • Map the clustering problem to a different domain
    and solve a related problem in that domain
  • Proximity matrix defines a weighted graph, where
    the nodes are the points being clustered, and the
    weighted edges represent the proximities between
    points
  • Clustering is equivalent to breaking the graph
    into connected components, one for each cluster.
  • Want to minimize the edge weight between clusters
    and maximize the edge weight within clusters

18
Characteristics of the Input Data Are Important
  • Type of proximity or density measure
  • Central to clustering
  • Sparseness
  • Dictates type of similarity
  • Adds to efficiency
  • Attribute type
  • Dictates type of similarity
  • Type of Data
  • Dictates type of similarity
  • Other characteristics, e.g., autocorrelation
  • Dimensionality
  • Noise and Outliers
  • Type of Distribution

19
Clustering Algorithms
  • K-means and its variants
  • Hierarchical clustering
  • Density-based clustering

20
K-means Clustering
  • Partitional clustering approach
  • Number of clusters, K, must be specified
  • Each cluster is associated with a centroid
    (center point)
  • Each point is assigned to the cluster with the
    closest centroid
  • The basic algorithm is very simple

21
Example of K-means Clustering
22
Example of K-means Clustering
23
K-means Clustering Details
  • Initial centroids are often chosen randomly.
  • Clusters produced vary from one run to another.
  • The centroid is (typically) the mean of the
    points in the cluster.
  • Closeness is measured by Euclidean distance,
    cosine similarity, correlation, etc.
  • K-means will converge for common similarity
    measures mentioned above.
  • Most of the convergence happens in the first few
    iterations.
  • Often the stopping condition is changed to Until
    relatively few points change clusters
  • Complexity is O( n K I d )
  • n number of points, K number of clusters, I
    number of iterations, d number of attributes

24
Evaluating K-means Clusters
  • Most common measure is Sum of Squared Error (SSE)
  • For each point, the error is the distance to the
    nearest cluster
  • To get SSE, we square these errors and sum them.
  • x is a data point in cluster Ci and mi is the
    representative point for cluster Ci
  • can show that mi corresponds to the center
    (mean) of the cluster
  • Given two sets of clusters, we prefer the one
    with the smallest error
  • One easy way to reduce SSE is to increase K, the
    number of clusters
  • A good clustering with smaller K can have a lower
    SSE than a poor clustering with higher K

25
Two different K-means Clusterings
Original Points
Sub-optimal Clustering
Optimal Clustering
26
Limitations of K-means
  • K-means has problems when clusters are of
    differing
  • Sizes
  • Densities
  • Non-globular shapes
  • K-means has problems when the data contains
    outliers.

27
Limitations of K-means Differing Sizes
K-means (3 Clusters)
Original Points
28
Limitations of K-means Differing Density
K-means (3 Clusters)
Original Points
29
Limitations of K-means Non-globular Shapes
Original Points
K-means (2 Clusters)
30
Overcoming K-means Limitations
Original Points K-means Clusters
One solution is to use many clusters. Find parts
of clusters, but need to put together.
31
Overcoming K-means Limitations
Original Points K-means Clusters
32
Overcoming K-means Limitations
Original Points K-means Clusters
33
Importance of Choosing Initial Centroids
34
Importance of Choosing Initial Centroids
35
Importance of Choosing Initial Centroids
36
Importance of Choosing Initial Centroids
37
Problems with Selecting Initial Points
  • If there are K real clusters then the chance of
    selecting one centroid from each cluster is
    small.
  • Chance is relatively small when K is large
  • If clusters are the same size, n, then
  • For example, if K 10, then probability
    10!/1010 0.00036
  • Sometimes the initial centroids will readjust
    themselves in right way, and sometimes they
    dont
  • Consider an example of five pairs of clusters

38
10 Clusters Example
Starting with two initial centroids in one
cluster of each pair of clusters
39
10 Clusters Example
Starting with two initial centroids in one
cluster of each pair of clusters
40
10 Clusters Example
Starting with some pairs of clusters having three
initial centroids, while other have only one.
41
10 Clusters Example
Starting with some pairs of clusters having three
initial centroids, while other have only one.
42
Solutions to Initial Centroids Problem
  • Multiple runs
  • Helps, but probability is not on your side
  • Sample and use hierarchical clustering to
    determine initial centroids
  • Select more than k initial centroids and then
    select among these initial centroids
  • Select most widely separated
  • Postprocessing
  • Bisecting K-means
  • Not as susceptible to initialization issues

43
Empty Clusters
  • K-means can yield empty clusters

Empty Cluster
44
Handling Empty Clusters
  • Basic K-means algorithm can yield empty clusters
  • Several strategies
  • Choose the point that contributes most to SSE
  • Choose a point from the cluster with the highest
    SSE
  • If there are several empty clusters, the above
    can be repeated several times.

45
Updating Centers Incrementally
  • In the basic K-means algorithm, centroids are
    updated after all points are assigned to a
    centroid
  • An alternative is to update the centroids after
    each assignment (incremental approach)
  • Each assignment updates zero or two centroids
  • More expensive
  • Introduces an order dependency
  • Never get an empty cluster
  • Can use weights to change the impact

46
Pre-processing and Post-processing
  • Pre-processing
  • Normalize the data
  • Eliminate outliers
  • Post-processing
  • Eliminate small clusters that may represent
    outliers
  • Split loose clusters, i.e., clusters with
    relatively high SSE
  • Merge clusters that are close and that have
    relatively low SSE
  • Can use these steps during the clustering process
  • ISODATA

47
Bisecting K-means
  • Bisecting K-means algorithm
  • Variant of K-means that can produce a partitional
    or a hierarchical clustering

48
Bisecting K-means Example
49
Hierarchical Clustering
  • Produces a set of nested clusters organized as a
    hierarchical tree
  • Can be visualized as a dendrogram
  • A tree like diagram that records the sequences of
    merges or splits

50
Strengths of Hierarchical Clustering
  • Do not have to assume any particular number of
    clusters
  • Any desired number of clusters can be obtained by
    cutting the dendrogram at the proper level
  • They may correspond to meaningful taxonomies
  • Example in biological sciences (e.g., animal
    kingdom, phylogeny reconstruction, )

51
Hierarchical Clustering
  • Two main types of hierarchical clustering
  • Agglomerative
  • Start with the points as individual clusters
  • At each step, merge the closest pair of clusters
    until only one cluster (or k clusters) left
  • Divisive
  • Start with one, all-inclusive cluster
  • At each step, split a cluster until each cluster
    contains a point (or there are k clusters)
  • Traditional hierarchical algorithms use a
    similarity or distance matrix
  • Merge or split one cluster at a time

52
Agglomerative Clustering Algorithm
  • More popular hierarchical clustering technique
  • Basic algorithm is straightforward
  • Compute the proximity matrix
  • Let each data point be a cluster
  • Repeat
  • Merge the two closest clusters
  • Update the proximity matrix
  • Until only a single cluster remains
  • Key operation is the computation of the proximity
    of two clusters
  • Different approaches to defining the distance
    between clusters distinguish the different
    algorithms

53
Starting Situation
  • Start with clusters of individual points and a
    proximity matrix

Proximity Matrix
54
Intermediate Situation
  • After some merging steps, we have some clusters

C3
C4
Proximity Matrix
C1
C5
C2
55
Intermediate Situation
  • We want to merge the two closest clusters (C2 and
    C5) and update the proximity matrix.

C3
C4
Proximity Matrix
C1
C5
C2
56
After Merging
  • The question is How do we update the proximity
    matrix?

C2 U C5
C1
C3
C4
?
C1
? ? ? ?
C2 U C5
C3
?
C3
C4
?
C4
Proximity Matrix
C1
C2 U C5
57
How to Define Inter-Cluster Distance
Similarity?
  • MIN
  • MAX
  • Group Average
  • Distance Between Centroids
  • Other methods driven by an objective function
  • Wards Method uses squared error

Proximity Matrix
58
How to Define Inter-Cluster Similarity
  • MIN
  • MAX
  • Group Average
  • Distance Between Centroids
  • Other methods driven by an objective function
  • Wards Method uses squared error

Proximity Matrix
59
How to Define Inter-Cluster Similarity
  • MIN
  • MAX
  • Group Average
  • Distance Between Centroids
  • Other methods driven by an objective function
  • Wards Method uses squared error

Proximity Matrix
60
How to Define Inter-Cluster Similarity
  • MIN
  • MAX
  • Group Average
  • Distance Between Centroids
  • Other methods driven by an objective function
  • Wards Method uses squared error

Proximity Matrix
61
How to Define Inter-Cluster Similarity
?
?
  • MIN
  • MAX
  • Group Average
  • Distance Between Centroids
  • Other methods driven by an objective function
  • Wards Method uses squared error

Proximity Matrix
62
MIN or Single Link
  • Proximity of two clusters is based on the two
    closest points in the different clusters
  • Determined by one pair of points, i.e., by one
    link in the proximity graph
  • Example

Distance Matrix
63
Hierarchical Clustering MIN
Nested Clusters
Dendrogram
64
Strength of MIN
Original Points
Six Clusters
  • Can handle non-elliptical shapes

65
Limitations of MIN
Two Clusters
Original Points
  • Sensitive to noise and outliers

Three Clusters
66
MAX or Complete Linkage
  • Proximity of two clusters is based on the two
    most distant points in the different clusters
  • Determined by all pairs of points in the two
    clusters

Distance Matrix
67
Hierarchical Clustering MAX
Nested Clusters
Dendrogram
68
Strength of MAX
Original Points
Two Clusters
  • Less susceptible to noise and outliers

69
Limitations of MAX
Original Points
Two Clusters
  • Tends to break large clusters
  • Biased towards globular clusters

70
Group Average
  • Proximity of two clusters is the average of
    pairwise proximity between points in the two
    clusters.
  • Need to use average connectivity for scalability
    since total proximity favors large clusters

Distance Matrix
71
Hierarchical Clustering Group Average
Nested Clusters
Dendrogram
72
Hierarchical Clustering Group Average
  • Compromise between Single and Complete Link
  • Strengths
  • Less susceptible to noise and outliers
  • Limitations
  • Biased towards globular clusters

73
Cluster Similarity Wards Method
  • Similarity of two clusters is based on the
    increase in squared error when two clusters are
    merged
  • Similar to group average if distance between
    points is distance squared
  • Less susceptible to noise and outliers
  • Biased towards globular clusters
  • Hierarchical analogue of K-means
  • Can be used to initialize K-means

74
Hierarchical Clustering Comparison
MIN
MAX
Wards Method
Group Average
75
MST Divisive Hierarchical Clustering
  • Build MST (Minimum Spanning Tree)
  • Start with a tree that consists of any point
  • In successive steps, look for the closest pair of
    points (p, q) such that one point (p) is in the
    current tree but the other (q) is not
  • Add q to the tree and put an edge between p and q

76
MST Divisive Hierarchical Clustering
  • Use MST for constructing hierarchy of clusters

77
Hierarchical Clustering Time and Space
requirements
  • O(N2) space since it uses the proximity matrix.
  • N is the number of points.
  • O(N3) time in many cases
  • There are N steps and at each step the size, N2,
    proximity matrix must be updated and searched
  • Complexity can be reduced to O(N2 log(N) ) time
    with some cleverness

78
Hierarchical Clustering Problems and Limitations
  • Once a decision is made to combine two clusters,
    it cannot be undone
  • No objective function is directly minimized
  • Different schemes have problems with one or more
    of the following
  • Sensitivity to noise and outliers
  • Difficulty handling different sized clusters and
    convex shapes
  • Breaking large clusters

79
DBSCAN
  • DBSCAN is a density-based algorithm.
  • Density number of points within a specified
    radius (Eps)
  • A point is a core point if it has more than a
    specified number of points (MinPts) within Eps
  • These are points that are at the interior of a
    cluster
  • A border point has fewer than MinPts within Eps,
    but is in the neighborhood of a core point
  • A noise point is any point that is not a core
    point or a border point.

80
DBSCAN Core, Border, and Noise Points
81
DBSCAN Algorithm
  • Eliminate noise points
  • Perform clustering on the remaining points

82
DBSCAN Core, Border and Noise Points
Original Points
Point types core, border and noise
Eps 10, MinPts 4
83
When DBSCAN Works Well
Original Points
  • Resistant to Noise
  • Can handle clusters of different shapes and sizes

84
When DBSCAN Does NOT Work Well
(MinPts4, Eps9.75).
Original Points
  • Varying densities
  • High-dimensional data

(MinPts4, Eps9.92)
85
DBSCAN Determining EPS and MinPts
  • Idea is that for points in a cluster, their kth
    nearest neighbors are at roughly the same
    distance
  • Noise points have the kth nearest neighbor at
    farther distance
  • So, plot sorted distance of every point to its
    kth nearest neighbor

86
Cluster Validity
  • For supervised classification we have a variety
    of measures to evaluate how good our model is
  • Accuracy, precision, recall
  • For cluster analysis, the analogous question is
    how to evaluate the goodness of the resulting
    clusters?
  • But clusters are in the eye of the beholder!
  • Then why do we want to evaluate them?
  • To avoid finding patterns in noise
  • To compare clustering algorithms
  • To compare two sets of clusters
  • To compare two clusters

87
Clusters found in Random Data
Random Points
88
Different Aspects of Cluster Validation
  • Determining the clustering tendency of a set of
    data, i.e., distinguishing whether non-random
    structure actually exists in the data.
  • Comparing the results of a cluster analysis to
    externally known results, e.g., to externally
    given class labels.
  • Evaluating how well the results of a cluster
    analysis fit the data without reference to
    external information.
  • - Use only the data
  • Comparing the results of two different sets of
    cluster analyses to determine which is better.
  • Determining the correct number of clusters.
  • For 2, 3, and 4, we can further distinguish
    whether we want to evaluate the entire clustering
    or just individual clusters.

89
Measures of Cluster Validity
  • Numerical measures that are applied to judge
    various aspects of cluster validity, are
    classified into the following three types.
  • External Index Used to measure the extent to
    which cluster labels match externally supplied
    class labels.
  • Entropy
  • Internal Index Used to measure the goodness of
    a clustering structure without respect to
    external information.
  • Sum of Squared Error (SSE)
  • Relative Index Used to compare two different
    clusterings or clusters.
  • Often an external or internal index is used for
    this function, e.g., SSE or entropy
  • Sometimes these are referred to as criteria
    instead of indices
  • However, sometimes criterion is the general
    strategy and index is the numerical measure that
    implements the criterion.

90
Measuring Cluster Validity Via Correlation
  • Two matrices
  • Proximity Matrix
  • Ideal Similarity Matrix
  • One row and one column for each data point
  • An entry is 1 if the associated pair of points
    belong to the same cluster
  • An entry is 0 if the associated pair of points
    belongs to different clusters
  • Compute the correlation between the two matrices
  • Since the matrices are symmetric, only the
    correlation between n(n-1) / 2 entries needs to
    be calculated.
  • High correlation indicates that points that
    belong to the same cluster are close to each
    other.
  • Not a good measure for some density or contiguity
    based clusters.

91
Measuring Cluster Validity Via Correlation
  • Correlation of ideal similarity and proximity
    matrices for the K-means clusterings of the
    following two data sets.

Corr -0.9235
Corr -0.5810
92
Using Similarity Matrix for Cluster Validation
  • Order the similarity matrix with respect to
    cluster labels and inspect visually.

93
Using Similarity Matrix for Cluster Validation
  • Clusters in random data are not so crisp

DBSCAN
94
Using Similarity Matrix for Cluster Validation
  • Clusters in random data are not so crisp

K-means
95
Using Similarity Matrix for Cluster Validation
  • Clusters in random data are not so crisp

Complete Link
96
Using Similarity Matrix for Cluster Validation
DBSCAN
97
Internal Measures SSE
  • Clusters in more complicated figures arent well
    separated
  • Internal Index Used to measure the goodness of
    a clustering structure without respect to
    external information
  • SSE
  • SSE is good for comparing two clusterings or two
    clusters (average SSE).
  • Can also be used to estimate the number of
    clusters

98
Internal Measures SSE
  • SSE curve for a more complicated data set

SSE of clusters found using K-means
99
Framework for Cluster Validity
  • Need a framework to interpret any measure.
  • For example, if our measure of evaluation has the
    value, 10, is that good, fair, or poor?
  • Statistics provide a framework for cluster
    validity
  • The more atypical a clustering result is, the
    more likely it represents valid structure in the
    data
  • Can compare the values of an index that result
    from random data or clusterings to those of a
    clustering result.
  • If the value of the index is unlikely, then the
    cluster results are valid
  • These approaches are more complicated and harder
    to understand.
  • For comparing the results of two different sets
    of cluster analyses, a framework is less
    necessary.
  • However, there is the question of whether the
    difference between two index values is
    significant

100
Statistical Framework for SSE
  • Example
  • Compare SSE of 0.005 against three clusters in
    random data
  • Histogram shows SSE of three clusters in 500 sets
    of random data points of size 100 distributed
    over the range 0.2 0.8 for x and y values

101
Statistical Framework for Correlation
  • Correlation of ideal similarity and proximity
    matrices for the K-means clusterings of the
    following two data sets.

Corr -0.9235
Corr -0.5810
102
Internal Measures Cohesion and Separation
  • Cluster Cohesion Measures how closely related
    are objects in a cluster
  • Example SSE
  • Cluster Separation Measure how distinct or
    well-separated a cluster is from other clusters
  • Example Squared Error
  • Cohesion is measured by the within cluster sum of
    squares (SSE)
  • Separation is measured by the between cluster sum
    of squares
  • Where Ci is the size of cluster i

103
Internal Measures Cohesion and Separation
  • Example SSE
  • BSS WSS constant

m
?
?
?
1
2
3
4
5
m1
m2
K1 cluster
K2 clusters
104
Internal Measures Cohesion and Separation
  • A proximity graph based approach can also be used
    for cohesion and separation.
  • Cluster cohesion is the sum of the weight of all
    links within a cluster.
  • Cluster separation is the sum of the weights
    between nodes in the cluster and nodes outside
    the cluster.

cohesion
separation
105
Internal Measures Silhouette Coefficient
  • Silhouette Coefficient combine ideas of both
    cohesion and separation, but for individual
    points, as well as clusters and clusterings
  • For an individual point, i
  • Calculate a average distance of i to the points
    in its cluster
  • Calculate b min (average distance of i to
    points in another cluster)
  • The silhouette coefficient for a point is then
    given by s (b a) / max(a,b)
  • Typically between 0 and 1.
  • The closer to 1 the better.
  • Can calculate the average silhouette coefficient
    for a cluster or a clustering

106
External Measures of Cluster Validity Entropy
and Purity
107
Final Comment on Cluster Validity
  • The validation of clustering structures is
    the most difficult and frustrating part of
    cluster analysis.
  • Without a strong effort in this direction,
    cluster analysis will remain a black art
    accessible only to those true believers who have
    experience and great courage.
  • Algorithms for Clustering Data, Jain and Dubes
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